Unveiling Weyl-related optical responses in semiconducting tellurium by mid-infrared circular photogalvanic effect

Elemental tellurium, conventionally recognized as a narrow bandgap semiconductor, has recently aroused research interests for exploiting Weyl physics. Chirality is a unique feature of Weyl cones and can support helicity-dependent photocurrent generation, known as circular photogalvanic effect. Here, we report circular photogalvanic effect with opposite signs at two different mid-infrared wavelengths which provides evidence of Weyl-related optical responses. These two different wavelengths correspond to two critical transitions relating to the bands of different Weyl cones and the sign of circular photogalvanic effect is determined by the chirality selection rules within certain Weyl cone and between two different Weyl cones. Further experimental evidences confirm the observed response is an intrinsic second-order process. With flexibly tunable bandgap and Fermi level, tellurium is established as an ideal semiconducting material to manipulate and explore chirality-related Weyl physics in both conduction and valence bands. These results are also directly applicable to helicity-sensitive optoelectronics devices.


Supplementary Section 1. Temperature-dependent resistance of tellurium
Temperature-dependent resistance of a typical tellurium (Te) device is shown in Supplementary Fig. 1, revealing typical characters of a doped semiconductor. As the temperature decreases, the resistance decreases first and then increases, reaching a minimum at T = 30 K. Below 30 K, phonon scattering no long dominates. The increase of resistance can be ascribed to the carrier freeze-out effect, which is a common behavior for semiconductors at low temperature.
Supplementary Fig. 1 Temperature-dependent resistance of Te.

Supplementary Section 2. Numerical simulations of CPGE responses
In this section, we use two different numerical approaches to show that sign reversal of CPGE at 4.0 μm and 10.6 μm excitation is consistent with optical selection rules determined by band structures in Te. In the numerical calculations of the both approaches, a tight-binding Hamiltonian is used to reproduce the electronic eigenstates, energy eigenvalues and to calculate velocity operators. The Hamiltonian is obtained from the band fitting to the first-principles energy bands with spin-orbital coupling by using the Wannier90 package 1

. The
, , three orbitals of Te are used in the band fitting, which are sufficient to make almost the same bands as the first-principles results.
Calculation of optical matrix elements is vital to determine the optical selection rules because the typical ones determined by two-band Hamiltonian of a single Weyl cone (as shown in Fig. 2a of the main text) fails in Te for the following three reasons. Firstly, both spin-splitting valence and conduction bands are formed by strong spin-orbit coupling in Te, so that spin angular momentum of specific energy band is not a good quantum number and cannot get a fixed value. Secondly, because the total angular momentum of photons and electrons must be conserved, orbital angular momentum has an impact on helicity-dependent optical selection rules. Large orbital angular momentum of electrons 2 on specific energy band must be considered when circularlypolarized light is incident to the system. Thirdly, transitions 2→3 and 2→4 cross the band gap and definitely goes beyond two-band Hamiltonian of a single Weyl cone. By calculating optical matrix elements for circularly polarized light, we can comprehensively consider factors above and directly obtain the probability of transitions caused by left or right circular polarization.

S 2.1. Optical matrix element of interband transitions between different energy bands of tellurium
In this section, we determine the helicity-dependent optical selection rules in Te by calculating optical matrix elements for interband transitions induced by circularly polarized light. Optical matrix elements can be calculated following equation S1: where | ( ) � is the electronic eigenfunction at momentum k of the valence (conduction) band; � denotes to velocity operator along direction and is given by: Considering that interband transitions are induced by the left and right circularly polarized light, and that the wave vector of light is along crystallographic y-axis in our experiment, the optical matrix elements should be written as equation S3: According to equation S3, we calculated optical matrix elements along the L2-H-L in Brillouin zone as shown in Supplementary Fig. 2  The above calculated results imply that, during interband transitions 1→2 and 2→3, electrons are always more likely to be generated with left-circularly polarized light while flow along the opposite directions. Subsequently, opposite CPGE is predicted to be obtained with 10.6-μm and 4.0-μm excitation, respectively. The calculated optical selection rules are shown in Fig. 2b of the main text, and sign reversal of CPGE with 10.6-μm and 4.0-μm excitation in our experiments is consistent with these selection rules. Additionally, 4.0-μm excitation may also induce optical transition from band 2 to 4 (2→4). Calculated optical matrix elements show little preference between the leftand right-circular polarizations along the L2-H-L ( Supplementary Fig. 2c). This result won't affect the consistency of calculations and experiments in our work.

S 2.2. Tensor elements of injection current
In this section, we verify the sign reversal of CPGE by calculating tensor elements of injection current with 10.6-μm and 4.0-μm excitations, respectively. This approach can provide the strength of CPGE quantitively. In the calculation, we also consider a cubic region rather than a single direction in Brillouin zone. As a result, this is a more direct and rigorous approach than calculating optical matrix elements.
Firstly, we determine the expression of the nonlinear photocurrent by symmetry analysis. Considering the symmetry of lattice, Te belongs to D3 point group. For optical electric field in x-z plane, nonzero tensor elements are and . The relationship between these two elements is: r and i denotes to the real and imaginary part of nonlinear tensor elements, respectively. Consequently, direct photocurrent generated from second order nonlinear effects can be obtained as: (0) = ( , − ) ( ) (− ) + ( , − ) ( ) (− ) + . . (S8) According to equation S4 and S7, the generated nonlinear photocurrent is determined by Secondly, we extract the CPGE responses from the total nonlinear photocurrent based on the dependence on the rotating angle of the quarter wave plate. Considering the fast axis of a quarter wave-plate is initially set along crystallographic a-axis and the wave plate is rotated continuously, the electric fields of the incident light follow: Here, 0 is the amplitude of incident light field, is the angle between the fast axis of quarter wave-plate and the crystallographic a-axis. Substitute and from equation S10 into equation S9, and the nonlinear photocurrent can be given by Considering that both left-and right-circular polarizations occur once when θ changes by 180 degrees, the CPGE responses we observed should correspond to the photocurrent component with a 180-degree period, namely 2 i 0 2 sin2 . The other photocurrent component has a 90-degree period, namely r 0 2 sin4 . Considering that the switch between a linear polarization and a circular polarization occurs once when θ changes by 90 degrees, the r term contributes the linear polarization dependent photocurrent response, while the CPGE responses are contributed by imaginary part of nonlinear photocurrent tensors only.
In a specific non-magnetic system such as Te, CPGE is generated by injection current effect, which is usually identified by an injection coefficient which relates to the conductivity as ∝ , and the injection coefficient is given by: Because this system is non-magnetic, we apply time reversal symmetry to both sides of equation S12 and find out that = − * , which means is a pure imaginary number. This is consistent with above analysis according to equation S11 in the previous paragraph and thus with our experiment. Namely, it is pure-imaginary injection coefficient that induces CPGE responses, which has a 180-degree period when the quarter wave-plate is rotated in our experiments.
Thirdly, we calculate the relevant tensor elements. Instead of numerically evaluating equation S12 directly, we consider the crystal symmetry and transform equation S12 into the following expression: Here, denotes to velocity matrix elements along direction k; a and b are index of energy bands, and c and v denotes to label empty and occupied states, respectively; denotes to the energy difference between the conduction and valence bands at specific position in the momentum space. Both and depend on momentum .
Considering the band edges appear at H1 and H2 points in the Brillouin zone, and they are connected by the transformation → − and → − as shown in Supplementary Fig. 3, the integrals in equation S13 near H1 and H2 are the same, so that we only need to calculate the imaginary part of the integral near a single H point. The integral is calculated in a small cubic of the momentum space with the H point as the center and 0.1 Å −1 as the side length. ( − ) is replaced with Gaussian , and the broadening Δ is set as 0.1 eV.

Supplementary Fig. 3 Brillouin zone of Te.
We first calculate tensor elements with 10.6-μm excitation. In this case, only transition  Supplementary Fig. 4a. Major peaks are observed at angular frequencies of 0, 1/π, and 2/π, which are attributed to polarization-independent, circular photogalvanic effect (CPGE), and anisotropic responses, respectively. The photocurrent components of different periodicities are plotted separately in Supplementary Fig. 4b. The CPGE response reaches local peaks under circular polarization excitation as marked by A and B. If we add polarization-independent, CPGE and anisotropic responses together, it recovers the experimentally measured signal plotted by the solid line in Fig. 2f of maintext. The peaks will shift from A and B towards A and B (the local peaks of 2/ π-periodicity component), leading to uneven distribution of photocurrent peaks. Fourier transform of photocurrent under 4.0-μm excitation in Fig. 2g of maintext is shown in Supplementary Fig. 4c and 4d.  In another measurement, another pair of electrodes, C and D, along the crystallographic a-axis, are connected for photocurrent measurement and the other four electrodes are floated. The spatial-resolved and polarization-dependent photocurrent responses are measured as shown in Supplementary Fig. 6. The CPGE responses show the same characteristics as those presented in the maintext.

S4.2. Results of device 3
For device 3, the CPGE responses also show the same characteristics as those presented in the maintext (Supplementary Fig. 7 and 8).   Fig. 3a and 3b of the maintext. The 4 ⁄ -dependent photocurrent measurement with different incident powers is also carried out on device 2, which is shown in Supplementary Fig. 10.

Supplementary Section 7. Alternative helicity-dependent photocurrent generation mechanisms
In this section, we discuss alternative helicity-dependent photocurrent generation mechanisms besides transverse CPGE presented in the maintext. We show that we can firmly rule out all other possibilities that may lead to helicity-dependent photocurrent response according to our measurement geometry and experimental results.

S7.1. Circular photon drag effect
In recent studies, circular photon drag effects (CPDE) are reported in Te and can produce transverse helicity-dependent photocurrent, too 3 . However, CPDE links to higher-order nonlinear tensor than CPGE in Te, which means a much smaller response magnitude. We note that third-order nonlinear optical tensor elements only contribute to CPGE, determined by * , while CPDE corresponds to higherorder nonlinear optical tensor elements, determined by * . Considering that rank-4 tensor is much smaller than rank-3 tensor, and is rather small for a photon, CPGE should contribute to much stronger (orders larger) signals on the generation of helicity-dependent photocurrent than that from CPDE.
Experimentally, PDE is usually observed in centrosymmetric materials where PGE is forbidden by inversion symmetry. In Te, CPDE can be observed only when a specific experimental geometry was selected carefully to exclude CPDE 3 . With experimental geometry in our work, both CPGE and CPDE can exist. Helicitydependent photocurrent can reach nearly 10 3 nA/W under 10.6-μm excitation in our work, while the maximum helicity-dependent photocurrent generated by CPDE is only around 10 nA/W with an incidence angle of 10 degrees according to previous measurements in the literature and is much smaller with an incidence angle of 0 degree 3 . So that helicity-dependent photocurrent should mainly originate from CPGE rather than CPDE, and the contribution from CPDE, even exists, is minor comparing to the response from CPGE in our work.

S7.2. Chiral edge currents
An edge photocurrent can stem from lower symmetry near the sample edges. Scattering of carriers driven by the circular polarized field determines the chirality of edge current. This phenomenon was first observed in graphene with THz radiation 4 . In our experiments, helicity-dependent photocurrent distributes at the area between two electrodes under 10.6-μm excitation, or near the sample-metal contact interfaces under 4.0-μm excitation, instead of the edges of samples. Edge currents should not play a role under the measurement geometry in our work.

S7.3. Photogalvanic effect driven by electron spin
Spin galvanic effect: helicity-dependent photocurrent was proposed and observed in semiconductor quantum well 5 driven by electron spin. Owing to spin-orbit interaction from asymmetric potentials, spin degeneracy of sub-bands can be lifted. Once these sub-bands are shifted in k space and inherent asymmetry in spin-flip transitions exists, a helicity-dependent photocurrent can be generated. However, this effect relies on combination of a magnetic field to realize rotation of non-equilibrium spin polarization.
Photo-induced inverse spin Hall effect: helicity-dependent photocurrent was observed in bulk GaAs due to photo-induced inverse spin hall effect 6 . In such effect, spin-dependent scattering happens during diffusion of photo-induced spin-oriented electrons from the surface into the bulk, resulting in observable helicity dependent photocurrent. However, this effect relies on a long spin-lifetime of electrons during the diffusion which does not apply to Te. The spin-degeneracy of conduction bands in GaAs leads to long spin-lifetime, while simultaneous lift of spin-degeneracy of both valence and conduction bands usually results in an extremely short spinlifetime 7 . We do not expect the photo-induced inverse spin Hall effect should play a vital role in such spin texture.

Supplementary Section 8. Fourier transform infrared spectroscopy of tellurium
Supplementary Fig. 12 Fourier transform infrared spectroscopy of Te. An unambiguous absorption edge cannot be extracted clearly but absorption behaviors can still be observed at 4.0 μm (0.31 eV) excitations.

Supplementary Section 9. Gate-voltage-dependent circular photogalvanic effect of tellurium
In this session, we measure the circular-polarization-dependent photocurrent under 4.0μm and 10.6-μm excitations by applying back-gate voltages to tune the doping of Te. When Fermi level is tuned by an electric gating, it may cause effects on optical transitions due to Pauli blockings. CPGE is expected to vanish or reverse under specific excitation wavelength and with suitable Fermi level. At the same time, the change of doping may affect the carrier densities involved in the optical transitions, which can affect the amplitude of photocurrent responses drastically.

S9.1. Gating effect by transport measurement at different temperatures
In a typical Te transistor, the hole-doping behaviors can be observed through the transfer curve measured at 1.5 K as shown in Supplementary Fig. 13. The chemical neutral point is obtained under a back-gate voltage at around 68.5 V. However, the current on/off ratio will go through a drastic drop and the chemical neutral point is shifted to a higher gate voltage when temperature increases as previously reported in the reference 8 . This is because thermal excitation leads to an increase of carrier densities at higher temperature, especially for narrow-bandgap materials. The thermal activation can provide available initial and final states for transition 1→2 under 10.6 μm and transition 2→3 under 4.0 μm, so that CPGE at specific back-gate voltages can be affected for room temperature measurement. Supplementary Fig. S13 Transfer curves of a typical Te transistor measured at 1.5 K.

S9.2. Gate-voltage-dependent circular photogalvanic effect under 4.0-μm excitation
In our experiment, 4.0-μm excitation can induce the spin-flip transition between two Weyl cones across the bandgap, from energy band 2 to 3 (2 → 3) as shown in Supplementary Fig. 14a. Ideally, applying a negative gate voltage will lower the Fermi level and lead to blocking of the transition 2→3 and emergence of the transition from energy band 1 to 2 (1→2) as shown in Supplementary Fig. 14a. CPGE should vanish at first, and then recover with a sign reverse.
The experiment is performed on device 4 and the Fermi level can be tuned by backgate voltages. Electrodes A and B along crystallographic a-axis are connected for photocurrent measurements and the other four electrodes are floated. Scanning photocurrent responses under 4.0-μm excitation are measured as shown in Supplementary Fig. 14b. The signs of photocurrent responses are opposite at two sample-metal contact interfaces.  Supplementary Fig. 14c and 14d. CPGE components extracted by Fourier transform are shown in Supplementary Fig. 14e and 14f Supplementary Fig. 15a. Ideally, applying a positive gate voltage will lift Fermi level and lead to blocking of the transition 1→2 and thus the related CPGE should disappear. In the measurement, electrodes C and D are connected for photocurrent measurements and the other four electrodes are floated. Scanning photocurrent responses under 10.6-μm excitation are measured as shown in Supplementary Fig. 15b. The photocurrent response mainly occurs near the electrode C.
4 ⁄ -dependent photocurrent responses are measured at the position with a maximal positive response on the device as shown in Supplementary Fig. 15c. The CPGE components extracted by Fourier transform are shown in Supplementary Fig. 15d. A back-gate voltage between ±106 V only changes the amplitude of CPGE while the expected vanishing of CPGE doesn't happen. Similar to the 4.0-μm excitation case, we speculate the thermal activation at room temperature should account for the CPGE at a high gate voltage by contributing available initial and final states for the transition 1→2 under 10.6-μm excitation. Other effects induced by the back gating beyond the modification of Fermi level can also have complicate contributions to the CPGE responses which remain to be studied in the future.